Find the largest number which divides $62$, $132$ and $237$ to leave the same remainder in each case.

### Concept & Logic To find the largest number that divides $x$, $y$, and $z$ leaving the same (but unspecified) remainder in each case, calculate the H.C.F. of the absolute differences between the given numbers. $$ \text{Required Number} = \text{H.C.F. of } |x - y|, |y - z|, \text{ and } |z - x| $$ ### Step-by-Step Solution * **Given:** The three numbers are $62$, $132$, and $237$. They must leave the same remainder when divided by the target number. * Find the differences between pairs of numbers: * $132 - 62 = 70$ * $237 - 132 = 105$ * $237 - 62 = 175$ * Now, calculate the H.C.F. of these differences: $70$, $105$, and $175$. * Factors of $70$: $1, 2, 5, 7, 10, 14, 35, 70$ * Factors of $105$: $1, 3, 5, 7, 15, 21, 35, 105$ * Factors of $175$: $1, 5, 7, 25, 35, 175$ * The highest common factor present in all three sets is $35$. ### Exam Strategy & Shortcut Use the **Difference Method for H.C.F.** on the new set of numbers ($70, 105, 175$). The difference between $105$ and $70$ is $35$. The H.C.F. must be $35$ or a factor of $35$. Check if $35$ divides all three differences: $70/35 = 2$, $105/35 = 3$, $175/35 = 5$. It divides them all, making $35$ the direct answer without writing out all factors. ### Common Pitfall Students often get stuck trying to set up algebraic equations with an unknown variable $R$ for the remainder (e.g., $62 = nX + R$). This wastes massive amounts of time. Immediately pivot to finding the differences of the numbers instead. ### Final Answer **Therefore, the correct answer is 35.**